= Pointwise limits in a functor category
{title2=$(\lim_jH_j)(d)=\lim_jH_j(d)$}
For small diagrams with complete codomain, choose each objectwise <categorical limit>. For $u:d\to d'$, its projections uniquely define the arrow $L(u)$ through $p_j(d')L(u)=H_j(u)p_j(d)$. The <universal property> proves identity and composition laws. The same projection calculation makes every <categorical cone> factorization a <natural transformation>. Hence these are genuine <categorical limits> in the <functor category>, and the objectwise <forgetful functor> is a <limit-creating functor>.
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