In comment #19 I remarked how \\(\mathrm{hom}\\) reminded me of a double-ended queue. Thinking about it some more, however, gave me the realization that there exist these six natural transformations,

\\[
\mathrm{AddFront}(e) := \\\\
\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h) \mapsto \mathrm{hom}(e,id\_{target(k)})\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h)
\\]

\\[
\mathrm{AddBack}(e) := \\\\
\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h) \mapsto \mathrm{hom}(id\_{source(j)},e)\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h)
\\]

\\[
\mathrm{DeleteFront} := \\\\
\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h) \mapsto \mathrm{hom}(id\_{target(j)},k)\cdots\mathrm{hom}(f,g)(h)
\\]

\\[
\mathrm{DeleteBack} := \\\\
\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h) \mapsto \mathrm{hom}(j,id\_{source(k)})\cdots\mathrm{hom}(f,g)(h)
\\]

\\[
\mathrm{PeekFront} := \\\\
\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h) \mapsto j
\\]

\\[
\mathrm{PeekBack} := \\\\
\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h) \mapsto k
\\]

Edit: Note that \\(\mathrm{hom}(j,k)\cdots\mathrm{hom}(f,g)(h)\\) is short for \\(j\circ \cdots \circ f \circ h \circ g \circ \cdots \circ k\\).