Let $fAb$ the abelian category of finite abelian groups, and let $\mathcal{C}:= Ind(fAb)$ its ind-category, this is the full category of presheaves on $fAb$ isomorphic to a filtred diagram of representable. Now form usual literature (e.g. Artin MAzur "Etale Homotopy" appendix) $\mathcal{C}$ is abelian, (then has finite sums), and has filtered colimits then has (small) sums, then is cocomplete.
Consider a countable numeration of finite cyclic groups $(C_n)_{n\in \mathbb{N}}$, and suppose that exist the product $P:= \prod_n h_{C_n}$ in $\mathcal{C}$ of associate representable of the $C_n$'s, let $P\cong \varinjlim_{i\in I} G_i$ h_{G_i}$for some direct diagrams of finite abelian groups$G_i$. . We have a split monomorphisms$\delta: \sum_n h_{C_n}\to P$, i claim the the family of maps$h_{C_n}\to \sum_n h_{C_n}\to P$is epimorphic, this follow because the the family$G_i\to P$is epimorphic, and any$G_j$is a (finite) sums of cyclic groups. But then$\delta$is a epimorphism, then a isomorphism. Now fix a cyclic group$C_{m}\neq 0$, and consider$(h_{C_m}, \sum_n h_{C_n})\cong \bigoplus_n fAb(C_m, C_n)$(the sum is a direct colimits of finite sums, and finite sums are representable by a biproduct) and each elemts of this sum as all$0$'s but finite components, but this isnt true for$(h_{C_m}, P)\cong \prod_n fAb(C_m, C_n)$, and considering that$1_{C_n}:= \pi_i\circ \delta\circ \epsilon_i : h_{C_n}\to \sum_n h_{C_n}\to P \to h_{C_n} $we get a absurd condition. 1 I'm no totally sure (as ever), If no I hope could suggest some ideas.. Let$fAb$the abelian category of finite abelian groups, and let$\mathcal{C}:= Ind(fAb)$its ind-category, this is the full category of presheaves on$fAb$isomorphic to a filtred diagram of representable. Now form usual literature (e.g. Artin MAzur "Etale Homotopy" appendix)$\mathcal{C}$is abelian, (then has finite sums), and has filtered colimits then has (small) sums, then is cocomplete. Consider a countable numeration of finite cyclic groups$(C_n)_{n\in \mathbb{N}}$, and suppose that exist the product$P:= \prod_n h_{C_n}$in$\mathcal{C}$of associate representable of the$C_n$'s, let$P\cong \varinjlim_{i\in I} G_i$for some direct diagrams of finite abelian groups$G_i$. . We have a split monomorphisms$\delta: \sum_n h_{C_n}\to P$, i claim the the family of maps$h_{C_n}\to \sum_n h_{C_n}\to P$is epimorphic, this follow because the the family$G_i\to P$is epimorphic, and any$G_j$is a (finite) sums of cyclic groups. But then$\delta$is a epimorphism, then a isomorphism. Now fix a cyclic group$C_{m}\neq 0$, and consider$(h_{C_m}, \sum_n h_{C_n})\cong \bigoplus_n fAb(C_m, C_n)$(the sum is a direct colimits of finite sums, and finite sums are representable by a biproduct) and each elemts of this sum as all$0$'s but finite components, but this isnt true for$(h_{C_m}, P)\cong \prod_n fAb(C_m, C_n)$, and considering that$1_{C_n}:= \pi_i\circ \delta\circ \epsilon_i : h_{C_n}\to \sum_n h_{C_n}\to P \to h_{C_n} \$ we get a absurd condition.