Suppose that $p_0:X_0\to K$ and $p_1:X_1\to K$ are two morphisms of simplicial sets and that $p_1$ is a (co)Cartesian fibration. We let ${\operatorname}{Fun}_{K}(X_0,X_1)$ denote the simplicial subset of all simplicial maps between $X_0$ and $X_1$ spanned by those maps which intertwine $p_0$ and $p_1$. Note that ${\operatorname}{Fun}_{K}(X_0,X_1)$ is automatically an $\infty$-category (cf. [@Lurie:2009un Remark 3.1.3.1]).

    When $p_0$ and $p_1$ are both (co)Cartesian fibrations, then we let $${\operatorname}{Fun}^{{\operatorname}{(co)Cart}}_K(X_0,X_1)\subseteq {\operatorname}{Fun}_{K}(X_0,X_1)$$ denote the subcategory spanned by those maps which preserve the (co)Cartesian edges.[^4]

Properties Inherited by Limit $\infty$-Categories
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Let ${{\mathscr{C}\mathrm{at}}}_\infty$ denote the $\infty$-category of small $\infty$-categories, let $K$ be a small simplicial set, and consider a diagram $p':K\to {{\mathscr{C}\mathrm{at}}}_\infty$. We will be interested in the limit $\infty$-category, $\lim p'\in {{\mathscr{C}\mathrm{at}}}_\infty$.

To compute such limits, consider the functor

$$\label{eq:DiagFunct}\Delta:{{\mathscr{C}\mathrm{at}}}_\infty\to {\operatorname}{Fun}(K,{{\mathscr{C}\mathrm{at}}}_\infty),$$

which sends an $\infty$-category ${\mathcal}{C}\in {{\mathscr{C}\mathrm{at}}}_\infty$ to the constant diagram: $$\Delta_{{\mathcal}{C}}:k\to {\mathcal}{C},\quad\text{ for any }k\in K.$$ The right adjoint to is the functor which sends a diagram $p'\in {\operatorname}{Fun}(K,{{\mathscr{C}\mathrm{at}}}_\infty)$ to the corresponding limit $\infty$-category $\lim p'\in {{\mathscr{C}\mathrm{at}}}_\infty$.

Now, fix a second diagram $p_0'\in {\operatorname}{Fun}(K,{{\mathscr{C}\mathrm{at}}}_\infty)$ and consider the functor $$\label{eq:DiagTensorFunctor}\Delta\times p_0':{{\mathscr{C}\mathrm{at}}}_\infty \to {\operatorname}{Fun}(K,{{\mathscr{C}\mathrm{at}}}_\infty),$$

which sends any $\infty$-category ${\mathcal}{C}\in {{\mathscr{C}\mathrm{at}}}_\infty$ to the functor $$\Delta_{\mathcal}{C}\times p_0':k\to {\mathcal}{C}\times p_0'(k),\quad\text{ for any }k\in K.$$ The right adjoint to 