This functor is a bit weird since the following holds where we can 'roll' everything to one side,

\\[
\begin{align}
\mathrm{hom}(l\circ f , g \circ j)(k) \\\\
= \mathrm{hom}(l\circ f,g)\circ\mathrm{hom}(k,j)(id\_{b'}) \\\\
= \mathrm{hom}(l\circ f,id\_{d'})\circ\mathrm{hom}(k,g)(j) \\\\
= \mathrm{hom}(l\circ f,id\_{d'})\circ\mathrm{hom}(j\circ k,id\_{d'})(g) \\\\
= \mathrm{hom}(k \circ l\circ f,id\_{d'})\circ\mathrm{hom}(g\circ j,id\_{d'})(id\_{d'}) \\\\
= \mathrm{hom}(g\circ j\circ k \circ l\circ f ,id\_{d'})(id\_{d'}),
\end{align}
\\]

and likewise to 'roll' everything in the other direction.