If we are dealing with the following reduction: $$\overline{K} \leq \{ p ∣ \forall y: M_p(y) \downarrow \} $$ one of the possible definitions for $M_p[x](y)$ is the following:
input y
{
if (not mxxstopsininputsteps)
accept;
else
infiniteloop;
}
But I cannot seem to wrap my head around the fact that we are just testing if $M_x(x)$ halts after $y$ steps (mxxstopswininputsteps). I have seen this method of construction of the mapping function in other more theoretical undecidability proofs (such as here: https://www.cs.rice.edu/~nakhleh/COMP481/final_review_sp06_sol.pdf) and I just do not understand how testing if it halts after $t$ steps is enough for concluding that $x \in \overline{K}$.