(I deleted my first attempt at an answer, as I had right and left reversed and anyway I wanted to try to say it better.)

I would advocate the following broad and comparatively low-tech view of the matter.

You are asking why two homotopy invariant versions of Hom of diagrams should be equivalent.

In general if you are dealing with a functor $F:C\to D$, where both $C$ and $D$ have some maps called weak equivalences, you can define the notion of "right derived functor" of $F$ as follows: Take the category of all functors $C\to D$. Make the slice category of all functors $(G,T:F\to G)$ under $F$. In it, consider the full subcategory whose objects are those $(G,T)$ such that $G$ preserves weak equivalences. In this category call a morphism $(G_1,T_1)\to (G_2,T_2)$ a weak equivalence if the map $G_1\to G_2$ is a weak equivalence (objectwise in $C$). By definition a right derived functor of $F$ is a homotopically initial object. That is, it is an object $(G,T)$ that becomes initial when you invert these equivalences.

From this definition it is clear that any two right derived functors of $F$ are equivalent, in the sense that there is zigzag of weak equivalences connecting them.

You care about the case when $C=Fun(A,Top)^{op}\times Fun(A, Top)$ and $F:C\to Top$ is the straightforward Hom functor, taking $(X,Y)$ to the set of natural maps between diagrams, topologized as a subspace of a product of function spaces, one for each $A$-object. Now you need to know why the various homotopy invariant replacements that you can think of for $Hom$ are in fact derived functors in the sense above.

The usual way of making a right derived functor for $F$ goes like this: think up a class $C_0$ of special $C$-objects such that $F$ takes equivalences between such objects to equivalences. Think up an endofunctor $r$ of $C$ such that

- $r$ takes every object to an object in $C_0$,
- there is a natural weak equivalence $e:1\to r$ to $r$ from the identity.
- $r$ takes weak equivalences to weak equivalences

As derived functor of $F$ use the composition $F\circ r$, equipped with the map $F\to F\circ r$ induced by $e$. This is in fact homotopically initial in that category of homotopy-invariant functors under $F$, under extremely weak axioms about $C$ and $D$ and their weak equivalences.

In your case you ought to be able to make $r$ and $e:1\to r$ by using cofibrant replacement of $X$ and fibrant replacement of $Y$ in either the projective or the injective model structure. The point to check, either way, is that maps $X_2\to X_1$ and $Y_1\to Y_2$ of diagrams induce equivalences $Hom(X_1,Y_1)\to Hom(X_2,Y_2)$ if the $X_i$ are cofibrant and the $Y_i$ are fibrant.