Suppose that we are in an ordinary ($\mathbf{Set}$-enriched) category $\mathcal{C}$. Is there a criterion that ensures the existence of finite tensors/finite cotensors? Does it suffice to be finitely bicomplete?
I just want to understand how these things that I am not familiar with behave...Do they behave like finite limits/colimits? For example, if a functor is left/right exact does it preserve finite tensors/cotensors?
How about a general $\mathcal{V}$-enriched category?
Regarding tensors: we're supposed to look for a universal property $\mathcal{C}(v\otimes c,c')\cong \mathcal{V}(v,[c,c'])$. When $\mathcal{V}=\textbf{Set}, v$ is a coproduct of points so $v\otimes c$ is just a coproduct of copies of $c$ indexed by $v$. In general, the given universal property is that of the weighted colimit $\{v,c\},$, where the objects $v,c$ are seen as functors out of the unit category $\mathcal{I}$. (If this isn't clear, observe that $\mathcal{V}$ is isomorphic to the functor category $[\mathcal{I},\mathcal{V}]$.) The story for cotensors is dual-they're limits of objects in $\mathcal{C}$ weighted by objects in $\mathcal{V}$.
If $\mathcal{C}$ is finitely bicomplete in the enriched sense, then it has tensors and cotensors by the above description. But if it's just bicomplete as a plain category, there are no such guarantees.