I have added (here) an observation that in a cohesive $\infty$-topos concrete objects are such that cohesive maps to them “glue”, in that every cohesive map out of a cover whose map of underlying $\infty$-groupoids descends down the cover also descends as a cohesive map.

This is an immediate consequence of the definitions and the (-1)-connected/(-1)-truncated orthogonality. The point is just to observe that this lifting problem does have this interpretation for concrete objects.

]]>the proof of that Proposition (still here) was lacking the argument that $im( \eta^{\sharp}_X )$ is indeed concrete. Have added that now.

]]>I have spelled out an argument (here) for the statement that in a local topos the full subcategory of concrete objects provides a factorization of $\Gamma \dashv coDisc$ as

$\Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set$ ]]>