When can we define the stalks of a sheaf?

Question

So i am reading rotman's book on homological algebra, and in it he states in Theorem 5.91, the following:
Let $X$ be a topological space and $\mathcal{A}$ an abelian category and consider the category of sheaves over $X$ with values in $\mathcal{A}$: $Sh(X,\mathcal{A})$. Then $Sh(X,\mathcal{A})$ itself is also an abelian category.
And in the proof he uses the stalks $\mathcal{F}_x$ of a sheaf $\mathcal{F}\in Sh(X,\mathcal{A})$. Rotman initially defined stalks only for the case $\mathcal{A}=Ab$, the category of abelian groups, which is fine, as this is cocomplete.
But in general $\mathcal{A}$ is not cocomplete, and we only know of the existence of finite colimits.
So my question is can we nonetheless define $\mathcal{F}_x$? Or do we need to suppose that $\mathcal{A}$ is cocomplete?