It's never too late to learn the following:
Let $\mathcal C$ be a $\mathcal V$-category; then, if $A\in\mathcal V$ is a left dualizable object (this means that $A$ regarded as an object of the one-object bicategory $\mathcal V$, has a left adjoint $^*\!A$), and the tensor $A\otimes C$ exists functorially, also the cotensor ${}^*\!A\pitchfork C$ exists functorially, and the two universal objects are isomorphic: $$ A\otimes C \cong\, {}^*\!A\pitchfork C $$ Furthermore, the colimit $A\otimes C$ is absolute.
I'm okay with this result, I can even cast it for more general shapes of weighted colimits. However, I am struggling to find an elementary proof of the last part: the colimit is absolute. Is there a conceptual, but explicit way, to prove it, that makes evident why it is true?