Morphism of k-schemes Let $k$ be an algebraically closed field. Let $X$ be a $k$-scheme.
I have been asked to "describe" the $k$-scheme morphisms of $\operatorname{Spec} k[x]/(x^2)$ to $X$.
I understand that $\operatorname{Spec} k$ is a one point set so the topological space level morphism has to be the constant function. How can I describe the sheaf level morphism? 
 A: I guess there are various possible answers that one can give here (with varying levels of sophistication). I am not sure that this answer is actively helpful to you, but maybe it is at least interesting:
We have the natural identification $$\text{Hom}_k(\text{Spec}(k[x]/(x^2)),X) = \text{Hom}_k(\text{Spec}(k),T_{X/k}) = T_{X/k}(k),$$ where $T_{X/k} = \text{Spec}(\text{Sym}(\Omega^1_{X/k}))$ is the  relative tangent bundle on $X$. This can be proven by direct computation locally in the affine case and then gives the above statement by gluing open affines. 
The observation above even gives another way to define the relative tangent bundle. The relative tangent bundle is the $k$-scheme that represents the presheaf $\mathcal{L}_1(X) \colon \text{Sch}/k^{op} \rightarrow \text{Set}$ of jets of level $1$ on $X$ defined by $$Y \mapsto \text{Hom}_k(Y \times_k k[x]/(x^{2}),X)$$ on objects and by sending a morphism $f \colon Y \rightarrow Y'$ to $g \mapsto g \circ \left(f \times_k \text{id}_{k[x]/(x^2)}\right)$.
