For a cardinal $\kappa$ let $[\kappa]^{<\kappa}$ denote the family of subsets of cardinality $<\kappa$ in $\kappa$. The family $[\kappa]^{<\kappa}$ is endowed with the partial order of inclusion. A simple diagonal argument shows that for any infinite cardinal $\kappa$ the poset $[\kappa]^{<\kappa}$ has cofinality $\ge\kappa$. It is easy to see that for a regular cardinal $\kappa$ this is an equality: the poset $[\kappa]^{<\kappa}$ has cofinality $\kappa$.

Problem.What is known about the cofinality of the poset $[\kappa]^{<\kappa}$ for a singular cardinal $\kappa$?Can it be equal to $\kappa$ for some special singular cardinals $\kappa$?

What is the cofinality of the poset $[\kappa]^{<\kappa}$ for $\kappa=\aleph_\omega$?

I hope that the answers should be known to specialists but I cannot find anything relevant in the books.

**Added in Edit:** In this answer @YCor proved that for any singular cardinal $\kappa$ the poset $[\kappa]^{<\kappa}$ has cofinality $>\kappa$. From this fact we can conclude that a cardinal $\kappa$ is regular if and only if $\mathrm{cof}([\kappa]^{<\kappa})=\kappa$. I suspect that this characterization (answering this MO-question) should be known. Does anybody know a suitable reference?