If $\mathsf{C}$ is a cocomplete category and $\mathsf{I}\to\mathsf{J}$ is a functor, when is $\mathsf{C}^\mathsf{J}\to\mathsf{C}^\mathsf{I}$ monadic?

Question

$\def\C{\mathsf{C}} \def\res{\operatorname{res}} \def\I{\mathsf{I}} \def\J{\mathsf{J}} \def\A{\mathsf{A}} \def\colim{\mathop{\operatorname{colim}}}$In Riehl's Category Theory in Context, we find:

Exercise 5.5.vi. Describe a general class of functors $K: \mathsf{I} \rightarrow \mathsf{J}$ between small categories so that for any cocomplete $\C$ the restriction functor $\operatorname{res}_K: \C^\mathsf{J} \rightarrow \C^\mathsf{I}$ strictly creates colimits of $\res_K$-split parallel pairs. All such functors admit left adjoints and are therefore monadic.

Right now I'm not interested in strict monadicity, so I'm just trying to find a class of functors $K$ for which $\res_K$ creates colimits of $\res_K$-split parallel pairs.

My guess is: If $\C$ is cocomplete, then $\res_K$ will satisfy this condition as long as $K$ is surjective on objects. I have two questions:

1. Do you find my proof below satisfactory? Or did I miss something?
2. Can you come up with alternative conditions on $K$ for which $\res_K$ becomes monadic?

Here's my proof: Suppose then $\C$ is cocomplete. Then $\C^\I$, $\C^\J$ are cocomplete as well. It suffices to show that $\res_K$ creates all colimits. Let $F:\A\to\C^\J$ be a diagram of functors (we will denote $F_a$ to the action of $F$ on the object $a\in\A$). The existence of $\colim\res_K\circ F$ trivially implies the existence of $\colim F$ (for the latter always exist). It is left to check that $\res_K$ reflects colimits. We will do so by application of this result. Here's when we use the condition that $K$ is surjective on objects: this implies that $\res_K$ is conservative (i.e., it reflects isos). It is left to see $\res_K$ preserves colimits. There's a canonical map $$ \tag{1}\label{can} \colim(\res_K F)\to\res_K(\colim F), $$ i.e., $$ \colim_{a\in\A}\underbrace{(\res_K F)_a}_{\res_KF_a}\to\res_K(\colim_{a\in\A} F_a). $$ Remember this is a morphism of functors living in $\C^\I$. Evaluated at $i\in\I$, the LHS reads: \begin{align*} \left(\colim_{a\in\A}\res_KF_a\right)(i) &=\colim_{a\in\A}\res_KF_a(i)\\ &=\colim_{a\in\A}F_aKi, \end{align*} whereas the RHS reads: \begin{align*} \res_K(\colim_{a\in\A}F_a)(i) &=\left(\colim_{a\in\A}F_a\right)(Ki)\\ &=\colim_{a\in\A}F_aKi. \end{align*} Since these are equal, the canonical morphism \eqref{can} must be an isomorphism. This means that $\res_K$ preserves all colimits.