Category Theory/(Co-)cones and (co-)limits

Definition (cone):

Let ${\mathcal {C}}$ be a category and let $G=(V,A)$ be a diagram in ${\mathcal {C}}$. A cone over $G$ is an object $a$ of ${\mathcal {C}}$, together with morphisms $\phi _{a,b}:a\to b$ for each $b\in V$, so that for each $\psi :b\to b'\in A$ (so that $b,b'\in V$) we have

$\phi _{a,b'}=\psi \circ \phi _{a,b}$.

Definition (limit):

Let ${\mathcal {C}}$ be a category, let $G=(V,A)$ be a diagram in ${\mathcal {C}}$ and let

Definition (diagonal):

Let ${\mathcal {C}}$ be a category, and let $X$ be an object of ${\mathcal {C}}$. Further, suppose that the object $X\times X$ and the morphisms $\pi _{1}:X\times X\to X$ and $\pi _{2}:X\times X\to X$ constitute a product of $X$ with itself in ${\mathcal {C}}$. Then the diagonal (also called "diagonal morphism") is the unique morphism

$\Delta :X\to X\times X$

such that $\pi _{1}\circ \Delta =\pi _{2}\circ \Delta =\operatorname {Id} _{X}$.

Definition (anti-diagonal):

Let ${\mathcal {C}}$ be a category, and let $X$ be an object of ${\mathcal {C}}$. Further, suppose that the object $X\sqcup X$ and the morphisms $\iota _{1}:X\to X\sqcup X$ and $\iota _{2}:X\to X\sqcup X$ constitute a coproduct of $X$ with itself in ${\mathcal {C}}$. Then the anti-diagonal (also called "anti-diagonal morphism") is the unique morphism

$\nabla :X\sqcup X\to X$

such that $\nabla \circ \iota _{1}=\nabla \circ \iota _{2}=\operatorname {Id} _{X}$.

Exercises

1. 1. Let ${\mathcal {D}}$ be a category such that every two objects of ${\mathcal {D}}$ have a product. Suppose further that ${\mathcal {C}}$ is another category, and that $T:{\mathcal {C}}\to {\mathcal {D}}$ is a functor. Let $Y$ be an object of ${\mathcal {D}}$. Use the universal property of the product in order to show that there exists a functor $S_{Y}:{\mathcal {C}}\to {\mathcal {D}}$ that sends an object $X$ of ${\mathcal {C}}$ to the object $T(X)\times Y$ of ${\mathcal {D}}$.
   2. Prove that any morphism $g:Y\to Z$ in ${\mathcal {D}}$ gives rise to a natural transformation $S_{Y}\to S_{Z}$.
   3. Can we weaken the assumption that every two objects of ${\mathcal {D}}$ have a product? (Hint: Consider the image of the class function on objects associated to the functor $T$.)