 a function of stacking angle, which are given in the Supplementary Information (Eq. S5). The SH dipole field is given by the product of d-tensors and the fundamental field, and the SHG intensity at the 1L MoS 2 /PZT interface is given by: … Figure 3f shows the simulated SHG results, which capture well the features of SHG tailoring effect for all stacking angles (Figs. 2e and 3e). Supplementary Table S2 lists a detailed comparison between the experimental and modeling results, which shows an excellent agreement, yielding strong support to the scenario for the interfacial polar coupling between MoS 2 and PZT DW." We also included the full modeling details, the explicit expressions of interfacial SHG tensors at the four DWs as a function of stacking angle, in the Supplementary Information, Section 5.

b) The authors should provide more detailed information on the SHG experimental setup and measurement conditions (e.g. spot size and focusing), in addition to what is currently provided on p.
12. This will also be useful to understand the SHG transmission results.

Reply:
We added detailed information of the experimental setup for the SHG measurement and the focusing condition for the fundamental beam to the Method section (Pages 16-17) as follows: "The laser beam passed a polarizer with normal incidence, and then was guided by mirrors into a laser scanning microscope (LSM). … The sample was place on a glass slide (1 mm), lying in the x-y plane, which is placed above the 1" diameter stage opening. The incident light was transversely polarized and directed to the sample surface along -z direction, and the excited SHG signals were collected in both reflection (+z) and transmission (-z) geometries by photomultiplier detectors (PMTs). … In the LSM, the transmission and reflection modes share the same focus plane using the same type of objective lens (NA1.05/water immersed/WD2.0 mm, 25×), so both measurements can be performed simultaneously. The focusing to the MoS 2 /PZT interface was performed through the reflected mode. The signal was first collected with no analyzer inserted, and then with an analyzer inserted in different angles with respect to the polarizer orientation. … The diffraction limit of the excitation laser beam (spot size) was estimated to be λ 2NA ⁄ 380 nm. Due to the second order nonlinearity of the SHG light, the spatial resolution was estimated to be ~300 nm." Comment 3.5: c) As done for PZT (Fig. S3), it would be good to provide measurements for SHG on MoS2-alone and MoS2 on +/-z PZT (as references), both for transmission and reflection. Current information provided by Fig.S5 is limited (and not very clear) in this respect.

Reply:
Following the reviewer's suggestion, we carried out SHG measurements on a MoS 2 sample (Fig. R8) as it was exfoliated on Gel-Film and after it was transferred on a PZT domain. As Gel-Film is a soft material, the SHG measurements were taken at a significantly lower laser power to avoid sample damage before transfer. For MoS 2 on Gel-Film, the sample exhibits similar SHG response in the reflection and transmission modes (Figs. R8c-d). For MoS 2 on PZT, we observed distinct SHG patterns in these two detection modes (Figs. R8g-h), similar to those shown in Figs. 2e and 4a. In the reflection mode, the P up and P down domains show comparable SHG response, while the DWs show a tailored pattern. In contrast, in the transmission mode, the P up and P down domains show distinct SHG intensities, while the signal at the DWs does not show appreciable contrast. We included this data in Supplementary Fig. S4, and revised the following sentences in the 1st paragraph on Page 5: "For both MoS 2 on Gel-Films and bare PZT, the SHG signals detected in the transmission mode exhibit qualitatively similar behavior as in the reflection mode ( Supplementary Fig. S3 and Fig. S4cd)."

Reply:
Following the reviewer's suggestion, we expanded Section 2 (Preparation and Characterization of Monolayer MoS 2 on PbZr 0.2 Ti 0.8 O 3 ) in the Supplementary Information. As shown in Fig. R8a-b, we first mechanically exfoliated MoS 2 flakes from a bulk single crystal on a Gel-Film (Fig. R8a), and identified the monolayer flakes via Raman measurements. For the selected flake, its crystalline orientation was identified by the polarized SHG measurement (Fig. R8b). We then created the P up and P down square domains on the PZT film. As shown in Figs. R8e-f, these PZT films have pre-deposited Au marks (50 m apart), which can help us locate the domain position and facilitate the alignment of the MoS 2 crystalline axis with the PZT DWs. During transfer, we aligned the a-axis (zigzag orientation) of MoS 2 with the horizental ([100]) DWs of PZT. After transfer, we used SHG mapping to check if we have achieved the desired alignment (Fig. R8g-h). During transfer, the uncertainty angle of alignment is less than 1°. There is, however, additional angular uncertainty during domain writing on PZT due to the thermal drift of the 21 AFM tip (up to 5°). So the overall uncertainty in the stacking angle is 2°-6°. We included this discussion in Supplementary Section 5, and added the following sentences in the first paragraph on page 16: "The uncertainty of stacking angle  is 2°-6°. The details of the sample alignment during transfer can be found in the Supplementary Information (Section 2 and Fig. S4)." Based on the estimated uncertainty, we updated Supplemental Table S2 as shown below. Overall, the simulated intensity distribution captures well the main features of the SHG tailoring pattern in the experimental results (Fig. 3e). Large deviation from the simulated values only occurs when the DW is in the vicinity of a cracked area in MoS 2, which compromises the net SHG signal.   Fig. 2a-h, Fig. 4a-c, Fig. S3a-h.

Reply:
The SHG mapping plots shown in the paper are the raw data collected by PMTs without modification using the same experimental setup, so the intensity levels can be directly compared if they were taken at the same excitation laser power. In the revised Fig. 4f (Fig. R7d), we compared the absolute SHG intensity on the P up and P down domains for both reflection and transmission modes.
As the PMTs were set on photon count mode during measurement, the SHG response is not calibrated to its corresponding light intensities in the units of W/m 2. We expressed the measured SHG strength in terms of arbitrary units (a.u.), which is proportional to the actual signal intensity measured by the PMTs. To clarify this point, we specified the excitation laser power either on the SHG image or in the figure caption, and added following sentence to the first paragraph on Page 18: "The SHG mapping plots, unless otherwise specified, are the raw data collected by the PMTs without modification. During the SHG measurements, the PMTs in the LSM were set on the photon count mode, so the responses of the PMTs are proportional to the number of actual SHG photons detected by the PMTs, but are not calibrated to the light intensity in units of W/m 2. The SHG mapping results were expressed in terms of arbitrary units (a.u.), and were proportional to the actual SHG intensity detected by the PMTs. The intensity level can be directly compared if they were taken at the same excitation laser power." Below are the comments in the attached pdf file.

Comment 3.8:
Page 3 -lines 19-20. When introducing the PZT material system, its crystallographic axes and the experimental reference frame should also be defined (this is done at some point in the Suppl. Information, but it is good to have this information at hand for all data. It would also eliminate possible ambiguities in using the terms 'parallel'and 'horizontal','vertical' in the text, see below)

Reply:
Following the reviewer's advice, we made the following revisions to the main text: 1. We defined the experimental reference frame and the crystalline orientation of PZT in Fig. 1 as insets, and labelled them wherever it fits. 2. We modified the following sentences in the last paragraph on Page 3: "The PZT films are (001)-oriented with out-of-plane polar axis ( Supplementary Fig. S1). Selected 1L MoS 2 flakes were transferred on top of the PZT thin film above a region patterned with a series of square domains with alternating up (P up

Reply:
We have compared the reflected and transmitted SHG responses of MoS 2 on Gel-Film and PZT P up and P down domains, as discussed in detail in the reply to Comment 3.5. Measurements on Gel-Films were taken at a significantly lower excitation power to avoid damage to the soft substrate (Figs. R7c-d). On Gel-Films, the SHG signal is similar in both detection modes. While measurements on standalone MoS 2 is not available, we do not expect there's any difference between the reflected and transmitted SHG. Fig. 2 and 3 fig. 2f. With the same methodology, looking at the d_xxx components