Applying Freyd-Mitchell's embedding theorem on large categories

One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category.

This is not immediately clear, since only small abelian categories can be embedded into R-mod.

Weibel for example argues that the snake lemma holds in an arbitrary abelian category (p. 12, Introduction to Homological Algebra):

The Snake Lemma also holds in an arbitrary abelian category $\mathcal{C}$. To see this, let $\mathcal{A}$ be the smallest abelian subcategory of $\mathcal{C}$ containing the
objects and morphisms of the diagram. Since $\mathcal{A}$ has a set of objects, the Freyd-Mitchell Embedding Theorem (see 1.6.1) gives an exact, fully faithful
embedding of A into R-mod for some ring $R$.

I am looking for a reference or an explanation as to why we know that the category $\mathcal{A}$, containing the diagram, has to be small.

I fear that this might be a stupid question, but why can't we potentially end up needing the whole category $\mathcal{C}$ to contain the diagram? I don't thoroughly understand how $\mathcal{A}$ will be constructed.

Thanks!

Just to get this out of the unanswered queue: yes, you can diagram chase even in a non-locally small abelian category using Freyd-Mitchell. Let $D:J\to \mathcal{A}$ be a (small) diagram in an arbitrary abelian category, and then work in the abelian category generated by $D(J)$. This is automatically small: you only need homs in the abelian groups generated by morphisms in $D(J)$, as well as by the kernels, cokernels, and finite direct sums of such morphisms. This process of taking kernels and generating morphism groups terminates after countably many steps, so in fact you get an abelian category of cardinality no more than $\aleph_0|J|$, where $|J|$ is the sum of cardinalities of the morphisms and objects of $J$.