An example of a category with all finite colimits and $\aleph_1$-filetered colimits but not all sequential colimits is the $\aleph_1$-filtered $Ind$-completion $Ind_{\omega_1}(\mathcal C)$ of any category $\mathcal C$ which doesn’t have sequential colimits.

A pretty natural example is the category of Banach spaces and all bounded linear maps (not just the contractive maps — if you take just the contractive maps you get a locally presentable category).

An example of a category with all finite colimits and $\aleph_1$-filetered colimits but not all sequential colimits is the $\aleph_1$-filtered $Ind$-completion $Ind_{\omega_1}(\mathcal C)$ of any category $\mathcal C$ which doesn’t have sequential colimits.

A pretty natural example of the example is the category of Banach spaces and all bounded linear maps (not just the contractive maps — if you take just the contractive maps you get a locally presentable category).

An example of a category with all finite colimits and $\aleph_1$-filetered colimits but not all sequential colimits is the $\aleph_1$-filtered $Ind$-completion $Ind_{\omega_1}(\mathcal C)$ of any category $\mathcal C$ which doesn’t have sequential colimits.

A pretty natural example of the example is the category of Banach spaces and all bounded linear maps (not just the contractive maps — if you take just the contractive maps you get a locally presentable category).

An example of a category with all finite colimits and $\aleph_1$-filetered colimits but not all sequential colimits is the $\aleph_1$-filtered $Ind$-completion $Ind_{\omega_1}(\mathcal C)$ of any category $\mathcal C$ which doesn’t have sequential colimits.

A pretty natural example is the category of Banach spaces and all bounded linear maps (not just the contractive maps — if you take just the contractive maps you get a locally presentable category).