Fix a simplicial set $\mathcal C$. We have a category of simplicial sets over $\mathcal C$, denoted by $\mathsf{sSet}/\mathcal C$. This is supposed to have an internal hom, denoted $\underline{\mathrm{Hom}}_{\mathcal C}(X, Y)$ for every two objects $X \to \mathcal C$ and $Y \to \mathcal C$ of $\mathsf{sSet}/\mathcal C$. How is this defined precisely? What are the $n$-simplices in $\underline{\mathrm{Hom}}_{\mathcal C}(X, Y)$?
What I know is that this should satisfy naturally the following equation, if $A \to \mathcal C$ is a third object of $\mathsf{sSet}/\mathcal C$:
$$ \mathrm{Hom}_{\mathsf{sSet}/\mathcal C}(A \times_{\mathcal C}X \to \mathcal C, Y\to \mathcal C) \cong \mathrm{Hom}_{\mathsf{sSet}/\mathcal C}(A \to \mathcal C, \underline{\mathrm{Hom}}_{\mathcal C}(X, Y)\to \mathcal C)$$
Can we use this in some way to identify the simplices of $\underline{\mathrm{Hom}}_{\mathcal C}(X, Y)$? Also, here, at page 208, this object is introduced, and it is claimed that morphisms $A \to \underline{\mathrm{Hom}}_{\mathcal C}(X, Y)$ over $\mathcal C$ correspond to morphisms $A \times_{\mathcal C} X \to A \times_{\mathcal C} Y$ over $A$. I think this is equivalent to the equality that I have written above, since it should be true that
$$\mathrm{Hom}_{\mathsf{sSet}/A}(A \times_{\mathcal C}X \to A, A \times_{\mathcal C} Y\to \mathcal A)\cong \mathrm{Hom}_{\mathsf{sSet}/\mathcal C}(A \times_{\mathcal C}X \to \mathcal C, Y\to \mathcal C),$$
but I am not sure, is this correct?