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A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.

Are homotopy fully faithful functors preserved under (homotopy) pushout?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.

Are homotopy fully faithful functors preserved under (homotopy) pushout?

More precisely, if $C\to D$ is fully faithful, and $C\to E$ is an arbitrary functor, is the canonical map $E\to E\sqcup_C D$ fully faithful?

A (homotopy) fully faithful functor is a map of $\infty$-categories which is essentially surjective and which induces weak equivalences on mapping spaces.

Are homotopy fully faithful functors preserved under (homotopy) pushout?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.

Are homotopy fully faithful functors preserved under (homotopy) pushout?