The category $PrL$ of locally presentable categories has all colimits. In particular, if $A_1 \leftarrow A_0 \rightarrow A_2$ is a diagram of presentable categories, with left adjoint functors between them, then the pushout $A_1 \coprod_{A_0} A_2$ in $PrL$ exists and the pushout functors $A_1 \rightarrow A_1 \coprod_{A_0} A_2, A_2 \rightarrow A_1 \coprod_{A_0} A_2$ are morphisms in $PrL$ and hence are left adjoints.

I am wondering whether this generalizes to categories that are not locally presentable. Concretely, if $A_1 \leftarrow A_0 \rightarrow A_2$ is a diagram of categories (maybe not presentable) with left adjoint functors between them, we can form the pushout $A_1 \coprod_{A_0} A_2$ in $Cat$.

**Question:** Are the pushout functors
$A_1 \rightarrow A_1 \coprod_{A_0} A_2$ and $A_2 \rightarrow A_1 \coprod_{A_0} A_2$ also left adjoints?

If the answer is no in general, are there additional conditions on the diagram $A_1 \leftarrow A_0 \rightarrow A_2$ that imply that the answer is yes?

This question Pushouts in the category of adjunctions is relevant but I think does not answer my question since I am not interested in pushouts in the category of adjunctions $Cat^{Adj}$, which I think does not have all pushouts by the answers there.

Remark: I am thinking about this question for $\infty$-categories and $Cat$ means that $\infty$-category of $\infty$-categories. It might be useful to consider $Cat$ as an $(\infty, 2)$-category but I am not sure.

generalizationof the fact in $Pr^L$ because the forgetful functor $Pr^L \to Cat$ does not preserve pushouts. $\endgroup$