It's well known that if $D\colon J \to \mathbf{Set}$ is a diagram where $J$ is a filtered category, and if $A$ is a finite set, then the natural map
$$ \text{colim}_{j}[A,D(j)] \to [A,\text{colim}_{j}D(j)] $$
is an isomorphism.
The definition of a filtered category is:
a) There is at least one object and for any pair $j,j'$ of objects, there is some $k$ such that there are morphisms $j\to k$, $j'\to k$.
b) For any morphisms $f,g\colon j'\to j$, there is some $h\colon j \to k$ such that $h\circ f=h\circ g$.
My question is about categories that satisfy (a) but not (b). I think (but am not completely sure) that the following is true.
If $J$ is a category satisfying property (a) then the map given above is a surjection.
Is this right? And is there a reference for it if so?
Update: attempted proof.
Let $f\colon A \to \text{colim}_jD(j)$ be a function. For each $a\in A$ choose some $j_a$ such that $f(a)\in D(j_a)$. Then, by inductively applying (a) to the $j_a$, we come up with some $k$ such that $f(a)\in D(k)$ for all $a$. Thus, $f$ may be written as a function $A \to D(k)$ and is therefore in the image of $\text{colim}_j[A,D(j)]$.