No a answere but a specification and description of the issue (too long for a comment).
The free cocompletion property of $Ind(\mathcal{C})$ dont use the hypothesis "$\mathcal{C}$ has finite colimits". ANyway $Ind(\mathcal{C})$ is equivalent to the full subcategory $P_1(\mathcal{C})\subset \mathcal{C}^>$ of presheves that are coimits of a small, filtred diagram of representable. In this way the inclusion $i$ corresponds to the yoneda inclusion $h_-: \mathcal{C}\to P_1(\mathcal{C})$ and it preserves limit, and finite colimits.The inclusion $P_1(\mathcal{C})\subset \mathcal{C}^>$ create (small) filter colimits (if $P=\varinjlim_{i\in I}P_i$ is a filtrant colimit with $O_i\in P_1(P_1(\mathcal{C})$ then combining the comma categories $P_1(\mathcal{C})\downarrow P_i$ wemake a filtrant (small) diagram of $P_1(\mathcal{C})\downarrow P_i$
Is $F: \mathcal{C}\to \mathcal{D}$ where $\mathcal{D}$ as filter colimits we have a (iso)unique extentions $F': P_1(\mathcal{C})\to \mathcal{D}$ with $F'(P):=\varinjlim_{\ (X, x)\in \mathcal{C}\downarrow P} F(X)$, if $P$ has the Ind-representation $(X_i)_{i\in I}$ i.e. $P=\varinjlim_i h_{X_i}$ then the diagram of the $h_{X_i}$ is a final diagram on the comma category $\mathcal{C}\downarrow P$ then $F'(P):=\varinjlim_i F(X_i)$
If $\mathcal{C}$ has finite colimits $P_1(\mathcal{C})\cong Cart(\mathcal{C}^{op}, Set)$ the latter is the the category cartesians presheaves i.e. that maps finite colimits of $\mathcal{C}$ to finite limits in $Set$, of course the embedding $Cart(\mathcal{C}^{op}, Set)\subset \mathcal{C}^>$ create limits .
All above as a dual version for $Proj(\mathcal{C}):=(Ind(\mathcal{C}^{op}))^{op}$, it is a free completion of $\mathcal{C}$, and if $\mathcal{C}$ has finite limits $Proj(\mathcal{C})$ is equivalent to $Cart(\mathcal{C}, Set)^{op}$ (dual to the category of copresheaves that preserving finite limits) it has (small) colimits and we have the embedding $\iota:=(h^-)^{op}: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op} $.
Now the inclusion $h_-: \mathcal{C}\to Cart(\mathcal{C}^{op}, Set)$ and $\iota: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op}$ induce for the universal properties of these two completions the functors $U: Cart(\mathcal{C}, Set)^{op}\to Cart(\mathcal{C}^{op}, Set)$ with $U(Q)=\varprojlim_{\ h^Y\to Q} h_Y$, and$F: Cart(\mathcal{C}^{op}, Set)\to Cart(\mathcal{C}, Set)^{op}$ with
$F(P)=\varprojlim_{\ h_X\to P} h^X$ (the limit is in $\mathcal{C}^{<}$ the copresheaves category)
we have the natural isomorphisms:
$Cart(\mathcal{C}, Set)^{op}(F(P), Q)=Cart(\mathcal{C}, Set)(Q, F(P))=$
$\mathcal{C}^{<}(\varinjlim_{\ h^Y\to Q}h^Y, \varprojlim_{\ h_X\to P}h^X)=$
$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^<(h^Y, h^X)\cong$
$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^>(h_X, h_Y)\cong$
$\mathcal{C}^>(\varinjlim_{\ h_X\to P} h_X, \varprojlim_{h^Y\to Q}h_Y) \cong$
$Cart(\mathcal{C}^{op}, Set)(P, U(Q))$
then $U$ is a adjoint to $F$.