Bijection induced by Hom and inner Hom

Question

For an $\infty$-category $X$ and a simplicial set $A\in sSet$. And for a category $\mathcal{C}$ we define $\mathcal{C}^\simeq$ whose objects are those of $\mathcal{C}$ and morphisms contain only isomorphisms in $\mathcal{C}$.
Moreover, we define $X\simeq$ for the $\infty$-category $X$ to be the pullback of the canonical map $X\to N(h_0(X))$ along the inclusion $N(h_0(X)^\simeq)\subseteq N(h_0(X))$. Here $h_0(X)$ denotes the homotopy category.
Define a subsimplicial set $K(A,X)\subseteq \underline{Hom}(A,X)$ by setting $K(A,X)_n=\{f:\Delta^n\times A\to X| \forall a\in A, f_a\in X_n^\simeq\}$
Define another subsimplicial set $h(B,X)\subseteq\underline{Hom}(B,X)$ by letting $h(B,X)=\{f:\Delta^n\times B\to X| \forall 0\leq i\leq n, \forall v:b_0\to b_1 \ in \ B, \ the \ map \ f(id,v):f(i,b_0)\to f(i,b_1) \ is \ invertible \ in \ X\}$
Now since we have the following isomorphisms:
$Hom(A,\underline{Hom}(B,X))\cong Hom(A\times B,X)\cong Hom(B\times A,X)\cong Hom(B,\underline{Hom}(A,X))$
Can we have a bijection:
$Hom(A,h(B,X))\cong Hom(B,K(A,X))$?
I have this spotted during a lecture note reading. The proof was skipped. I have a hard time seeing the bijection. Any help is appreciated.