Does the assignment sending every topological space to its sheaf hom functor a functor?

Question

Let $X$ be a topological space. Given sheaves $F,G\in\text{Sh}(X)$, the sheaf hom of $F$ and $G$ is the sheaf $\text{Hom}_X(F,G)$ of $X$ defined on objects by $$\text{Hom}_X(F,G)(U)=\text{Nat}(F|_U,G|_U)$$ (where $F|_U$ is the restriction of $F$ to $U$). This in turn defines a functor $$\text{Hom}_X:\text{Sh}(X)^\mathrm{op}\times \text{Sh}(X)\to \text{Sh}(X)$$ What I want to know is whether if the assignment $X\mapsto \text{Hom}_X$ defines yet another functor $$\mathbf{Top}\to [\text{Sh}(-)^\mathrm{op}\times \text{Sh}(-), \text{Sh}(-)]$$ (I don't think the rightmost notation is formal enough but should be evident on what I'm trying to say).