Let $I$ be a small category and $\mathcal{C}$ an $I$-complete category. Denote $\iota : I \rightarrow \hat I$ the inclusion into the category obtained from $I$ by “adding an initial object”. A cone over some diagram $D: I \to \mathcal{C}$ can be considered as a diagram $\hat D : \hat I \to \mathcal{C}$ such that $[\iota, \mathcal{C}](\hat D) = D$, where $[ \iota, \mathcal{C}] : [\hat I,\mathcal{C}] \to [I, \mathcal{C}]$ is the obvious functor associated to $\iota$.

Now it is easy to see that the functor $lim_I :[I, \mathcal{C}] \to [\hat I,\mathcal{C}]$ associating to each diagram its limit-cone is a right adjoint of $[ \iota, \mathcal{C}] : [\hat I,\mathcal{C}] \to [I, \mathcal{C}]$ (the counit is just the identity). Thus, $lim_I$ could be actually defined in this way up to isomorphism (because of unicity of adjoints up to isomorphism).

My question is wether this works even if we don't require $\mathcal{C}$ to be $I$-complete, more precisely: is any category $\mathcal{C}$, for which $[ \iota, \mathcal{C}] : [\hat I,\mathcal{C}] \to [I, \mathcal{C}]$ has a right adjoint, $I$-complete (so that we can then define $lim_I$ to be this right adjoint)?

It seems to me that we might have to require the counit of the adjunction to be an isomorphism. In that case $-$ have such adjunctions been studied (and if yes, where can I read about it)? are they closed under composition for example?

Note that the diagonal functor $\Delta\colon \mathcal C \to [I,\mathcal C]$ factorizes as $\mathcal C \to [\hat I,\mathcal C]\to [I,\mathcal C]$ where the first functor is the diagonal functor for $\hat I$ and the second functor is the restriction functor you describe. Further, the right adjoint $[\hat I,\mathcal C]\to \mathcal C$ certainly exists, since $\hat I$ has an initial object. So, if a right adjoint $[I,\mathcal C]\to [\hat I,\mathcal C]$ exists, then the composition $[I,\mathcal C]\to [\hat I,\mathcal C]\to C$, which maps a diagram to its limiting object, must be a right adjoint to the diagonal functor $\mathcal C \to [I,\mathcal C]$, and so is, up to isomorphism, the limits of shape $I$ functor.