Functors between structured categories

The pair $(\mathfrak K,P)$ consisting of a category $\mathfrak K$ and a univalent functor $P$ from $\mathfrak K$ to a category $\mathfrak U$ is called a structured category. If $(\mathfrak K_1, P_1)$ and $(\mathfrak K_2,P_2)$ are two such pairs, then a functor $F\colon\mathfrak K_1\to\mathfrak K_2$ is structured if $FP_2=P_1$. Conditions are determined under which all structured functors have a left adjoint functor.
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