I have drawn the picture like the following.
Let $C$ is a category (suppose $Set$), $I$ is a category with only two objects (1 and 2), $\Delta_{A\times A}$ and $D$ are constant functors from $I$ to C.
A cone on $D$ with a vertex $u$ is a natural transformation from $\Delta_{u}$ to $D$. A limit of D is a universal cone, that is for any cone on $D$ there exists a unique factorization.
Obviously the product $A \times A$ equipped with two projections $fst$ and $snd$ is a limit of $D$.
My question is that how many universal cones (or limits) on $D$?
On the surface there seems to be only one universal cone, but I thinks there are two.
The reason is that a cone is essentially a natural transformation that satisfies some commuting conditions.
In Mac Lane p.16, he said that
a natural transformation is a function... which assigns to each object c of C an arrow $τ_{c} = τc : S_c \to T_{c}$ of B in such a way ...
There are two such natural transformations, i.e. $\alpha$ and $\beta$.
$\alpha: \Delta_{A\times A} \to D$ with 2 components
$\alpha_{1} = fst: A \times A \to A$
$\alpha_{2} = snd: A \times A \to A$
$\beta: \Delta_{A\times A} \to D$ with 2 components
$\beta_{1} = snd: A \times A \to A$
$\beta_{2} = fst: A \times A \to A$
Here, the $\alpha$ and $\beta$ are obviously different functions.
Am I correct?
Thanks.
For clarity, I've slightly edited the question, but the overall point remains unchanged.
An immediate consequence of the universal property is that between any two universal cones $(\alpha_1,\alpha_2)$ and $(\beta_1,\beta_2)$ there are unique morphisms $b$ and $a$ for which $b\circ\alpha_i=\beta_i$ and $a\circ\beta_i=\alpha_i$. But then $a\circ b\circ\alpha_i=a\circ\beta_i=\alpha_i$ and $b\circ a\circ\beta_i=b\circ\alpha_i=a_i$. Another application of the universal property then yields that $a\circ b=\mathrm{id}$ and $b\circ a=\mathrm{id}$, i.e. that $a$ and $b$ are inverse isomorphisms.
Thus the answer to the question is that if there is at least one universal cone, say $\beta_1,\beta_2\colon B\rightrightarrows A$, then you have as many universal cones as isomorphisms with domain $B$.
Of particular interest to you seem to be other universal cocones with the same vertex $B$, which would correspond to automorphisms of $B$, i.e. isomorphisms of $B$ with itself. These in general depend on the category, but certain of the isomorphisms do not, i.e. those constructed from the projection morphisms alone.
Explcitly, the univeral property implies that endomorphism $B\to B$ correspond to cones $B\rightrightarrows A$, and there are exactly four that can be constructed from the two projections $\beta_1,\beta_2\rightrightarrows A$: $(\beta_1,\beta_2)$, $(\beta_1,\beta_1)$, $(\beta_2,\beta_2)$, and $(\beta_2,\beta_1)$. These correspond to the identity, left, right, and reverse endomorphisms of $B$. The identity and reverse endomorphisms are the only isomorphisms. Thus every universal cone has exactly one other universal cone structure that exists independently of the ambient category.