How to show the $\kappa$-small functor is $\kappa$-accessible? (coalgebraic logic)

Question

A $\mathtt {Set}$-functor $T:\mathtt {Set} \to \mathtt {Set}$ is defined to be $\kappa$-accessible for a regular cardinal $\kappa$ iff for all sets $X$ and all $x\in TX$ there exists a subset $Y\subseteq X$ with $|Y|<\kappa$, such that $x\in T_i (Y)$, where $i:Y \to X$ is the inclusion function.

Now some resources define a $\kappa$-small functor $T_\kappa$, obtained from an arbitrary $\mathtt {Set}$-functor $T$, to be defined as follows:

$T_\kappa (X):=\cup\{T_i(TA) | i:A <\kappa\}$, and

$T_\kappa (f):=T(f)$.

Now we want to show that the functor $T_\kappa$ is $\kappa$-accessible. I couldn't do this. May you please help on this?

Thanks.

Answer 1

Well I found that the problem is with the definition of $\kappa$-accessibility and many resources just don't care about this notational problem and take it for granted! Here we go:

A $\mathtt {Set}$-functor $T:\mathtt {Set} \to \mathtt {Set}$ is defined to be $\kappa$-accessible for a regular cardinal $\kappa$ iff for all sets $X$ and all $x\in TX$ there exists a subset $Y\subseteq X$ with $|Y|<\kappa$, such that $x\in T_i (T[Y])$, where $i:Y \to X$ is the inclusion function.

Then every thing will be open.

PS: $T_i (Y)$ makes no sense in general, because we have $T_i:TY \to TX$ and it's not guaranteed that $Y \subseteq TY$.