Adjoint functors and free objects

Question

Let $L$ and $C$ be a couple of adjoint functors between two categories $A$ and $B$ that both have free objects. If $F$ is free in $A$, would $L(F)$ be free in $B$?

Answer 1

If you mean what Captain Lama wrote then yes this is true if we include that the triangle of forgetful functors has to commute.

Where $U_A$ and $U_B$ are the two fixed forgetful functors whose left adjoints are denoted $F_A$ and $F_B$ respectively.

This is due to the fact that if we have two left/right adjoint pairs $(L_1, R_1)$ and $(L_2, R_2)$ $$X \xrightarrow {R_1} Y \xrightarrow {R_2} Z $$ then $L_1L_2$ is left adjoint to $R_2R_1$.

In your case you can take $R_1 = C$ and $R_2 = U_B$, then $L_1L_2 = L F_B$ is left adjoint to $R_2R_1 = U_A$.

But $F_A$ is also a left adjoint to $U_A$ so by the uniqueness of adjoints we have to have $LF_B \approx F_A$ (natural isomorphism). This means that applying $L$ to a free object in $B$ yields a free object in $A$.

More precisely, if $X \in B$ is freely generated by a set $S$ then $LX \in A$ is freely generated by the same set $S$.

Note also that $U_A$, $U_B$ and $C$ dont have to be faithful, this applies to any functors with left adjoints.