Lemma 4.22.8. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered (resp. cofiltered) index category $\mathcal{I}$. Let $F : \mathcal{C} \to \mathcal{D}$ be a functor. If $M$ is essentially constant as an ind-object (resp. pro-object), then so is $F \circ M : \mathcal{I} \to \mathcal{D}$.

**Proof.**
If $X$ is a value for $M$, then it follows immediately from the definition that $F(X)$ is a value for $F \circ M$.
$\square$

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