that this map $\Theta$ is well-defined and that it is actually bijective.

The action of Kashiwara operators on $\M(\infty)$ given in Lemma \[anaysiskashi2\] follows the process for defining it on $\T(\infty)$. Hence, the map $\Theta$ naturally commutes with the Kashiwara operators $\fit$ and $\eit$. Other parts of the proof are similar or easy.

\[final2\] From Proposition \[cor4\] and Theorem \[mainreal5\], we can conclude that the crystal $\M(p_1,p_2;r;\infty)$ is also $\uq(G_2)$-crystal isomorphic to $\B(\infty)$.

We illustrate the correspondence between $\T(\infty)$ and $\M(\infty)$. A monomial of $\M(\infty)$ $$\begin{aligned}
M=& {Y_1(-1)}^{(1,1)}{Y_1(1)}^{(0,-5)}{Y_1(2)}^{(0,-1)}\\
  &\cdot {Y_2(-2)}^{(1,-2)}{Y_2(-1)}^{(0,-1)}{Y_2(0)}^{(0,2)}\end{aligned}$$ can be expressed as $$\begin{aligned}
M=& {X_1(-1)}^{(2,-5)}
    {X_2(-1)}^{(0,1)}
    {X_0(-1)}^{(0,1)}
    {X_{\bar 3}(-1)}^{(0,2)}
    {X_{\bar 1}(-1)}^{(0,1)}\\
  &\cdot {X_2(-2)}^{(1,-2)}
         {X_3(-2)}^{(0,2)}\end{aligned}$$ by (\[equ:jin19\]). Hence we have the following marginally large tableau as the image of $M$ under $\Theta^{-1}$. $$\Theta^{-1}(M)=
\raisebox{-1.3em}{
\begin{texdraw}\drawdim in
\arrowheadsize l:0.065 w:0.03
\arrowheadtype t:F
\fontsize{5}{5}\selectfont
\textref h:C v:C
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\setunitscale 1.6
\move(0 2)\rlvec(9 0)
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\move(0 0)\rlvec(3 0)\rlvec(0 2)
\move(0 2)\rlvec(0 -2)
\move(1 0)\rlvec(0 2)
\move(2 0)\rlvec(0 2)
\move(4 1)\rlvec(0 1)
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\htext(0.5 1.5){$1$}
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\htext(6.5 1.5){$\bar 3$}
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\htext(0.5 0.5){$2$}
\htext(1.5 0.5){$3$}
