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For a diagram $D:\mathcal{J}\to \mathcal{C}$, a limit-preserving functor carries every limiting cone over a diagram to a limiting categorical cone. A limit-reflecting functor has the converse property: a categorical cone is limiting whenever its image is limiting. A limit-creating functor uniquely lifts every specified limiting categorical cone over the image diagram to a categorical cone over $D$, and the lift is limiting. These definitions concern diagrams of the stipulated shape; creation includes the lifting requirement, not merely reflection.

Let $(L,(p_j))$ be a categorical limit of $D$. The mapis a bijection: the right side consists exactly of compatible families of arrows from $A$, and the categorical limit's universal property gives their unique factorization through $L$. The bijection is induced by the categorical limit projections, so it proves that covariant representables preserve limits, including the empty diagram and its terminal object.

Construct pointwise limits in a functor category. For $H:\mathcal{J}\to [\mathcal{D},\mathcal{C}]$, choose at each object $d$ a categorical limit $L(d)$ of $j\mapsto H_j(d)$, with projections $p_j(d)$. For $u:d\to d^{\prime }$, the family $H_j(u)p_j(d)$ is compatible; define $L(u)$ uniquely byThe universal property gives $L(1_d)=1_{L(d)}$ and $L(vu)=L(v)L(u)$, since those equalities hold after every projection. Thus $L$ is a functor, and each $p_j:L\Rightarrow H_j$ is a natural transformation.

For any categorical cone $(M,(a_j))$, the pointwise categorical limits give unique maps $a(d):M(d)\to L(d)$. To check naturality, compose $L(u)a(d)$ and $a(d^{\prime })M(u)$ with every $p_j(d^{\prime })$; both become $H_j(u)a_j(d)=a_j(d^{\prime })M(u)$. The projections distinguish arrows into their categorical limit, so the two maps agree. Componentwise uniqueness gives uniqueness of the natural transformation $a$. This proves that $(L,p_j)$ really is the required categorical limit, rather than merely a family of objectwise candidates.

A specified categorical limit categorical cone in ${\mathcal{C}}^{\mathrm{ob}\mathcal{D}}$ fixes these objectwise vertices and projections. The displayed equation forces every arrow $L(u)$, and the argument forces every categorical cone factorization. Hence the forgetful functor uniquely lifts that categorical cone and is a limit-creating functor. The construction only takes small categorical limits in $\mathcal{C}$; it does not require $\mathcal{D}$ to be small. As usual, the functor categories are understood in a universe where their collections of transformations are meaningful.

For small diagrams with complete codomain, choose each objectwise categorical limit. For $u:d\to d^{\prime }$, its projections uniquely define the arrow $L(u)$ through $p_j(d^{\prime })L(u)=H_j(u)p_j(d)$. 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.