I'm looking at a justification for why we work inside a big model, but I'm having trouble proving a particular comment.
Assume we're working with a complete theory $T$ and we have a commutative diagram of models of $T$ with the maps being elementary embeddings. I'm trying to show that I can find a copy of this diagram inside a larger model $\mathcal{M}$ of $T$, where the embedding maps become inclusions and the models in the diagram are elementary submodels of $\mathcal{M}$.
I know we can take any two models of $T$ and embed them into some other model of $T$, but I can't figure out how to embed the models in a commutative diagram so that the embedding maps become inclusions. Any help is appreciated.
This is just another instance of the "Valby motto" of compactness: Write down what you want, and get what you want!
Let $\{M_i\mid i\in I\}$ be the models appearing in the diagram. For all $i\in I$, let $\mathrm{Diag}(M_i)$ be the elementary diagram of $M_i$ in the language $L_i$ expanded by constant symbols $\{a_i\mid a\in M_i\}$. Let $L'$ be the language $\bigcup_{i\in I} L_i$, and let $T'$ be the $L'$-theory $$\bigcup_{i\in I} \mathrm{Diag}(M_i)\cup \{a_i = b_j\mid f(a_i) = b_j\text{ for some map }f\colon M_i\to M_j\text{ in the diagram}\}.$$ I'll leave it to you to show that $T'$ is consistent by compactness, using the fact that the diagram commutes and all the maps are elementary embeddings.
Now let $\mathcal{M}$ be a model of $T'$. Identifying each $M_i$ with the set of interpretations of its constants in $\mathcal{M}$, each $M_i$ is an elementary submodel of $\mathcal{M}$ (since $\mathcal{M}\models \mathrm{Diag}(M_i)$) and the maps in the diagram are inclusions (since $a_i = b_j$ when $f(a_i) = b_j$ for some map in the diagram).
This is a generalization of the fact that the models of a complete theory have the amalgamation property. If you have trouble verifying the compactness argument above, see Theorem 5.3.1 (Elementary amalgamation theorem) in Hodges' A Shorter Model Theory for a detailed proof of the easier fact, which may help you work out the details of the more general fact.