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\$$.