Let $\mathcal{C}$ and $\mathcal{D}$ be categories, $\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$ be a functor. There is a notion of limit of $\mathcal{F}$, namely a pair $(\ell,\varphi)$, where $\ell\in\text{Obj}(\mathcal{D})$ is an object and $\varphi\colon\Delta_{\ell}\to\mathcal{F}$ is a natural transformation from the constant functor to $\mathcal{F}$, such that for any other pair $(d,\alpha)$ of the same type (which is called a cone) there exists a unique morphism $f\colon d\to\ell$, such that $\varphi\circ\Delta_f=\alpha$.
Obviously the concept of limit is a conjunction of the two properties:
1). for every cone $(d,\alpha)$ there exists a morphism $f\colon d\to\ell$, such that $\varphi\circ\Delta_f=\alpha$;
2). if such a morphism exists, then it is unique.
One may be interested what happens if we reject one of these requirements. For instance, if we reject the uniqueness condition, then we come to the concept of weak limit.
Question: What concept do we get if we reject the existence condition?
Of course, we may define it and call it a soft limit. We also may find some examples (empty set is a soft terminal object of $\mathbf{Set}$). My question is whether it was studied anywhere or if this is a special case of another known category-theoreric concept.
What you call a "soft terminal object" is known as a subterminal object; it can be characterized as an object such that the two projections $X\times X\to X$ and the diagonal $X\to X\times X$ are isomorphisms, or equivalently as a subobject of the terminal object, if there is one.
More generally, a family of morphisms $\alpha_i:A\to X_i$ for $i\in I$ such that $\alpha_i\circ f=\alpha_i\circ g$ for all $i$ implies $f=g$ is called a jointly monic family (or mono-source), and this condition is equivalent to the canonical map $A\to \prod_i X_i$ being a monomorphism (if the product exists). So I guess you could call your "soft limits" sublimits, or jointly monic cones.