Sugar – Exact proportions of sugar to water in sorbet

The problem could be the recipe. If you're using a lot of lemon juice, then a lot of sugar to cover up all that acidity, you're guaranteed to end up with something pretty soft. Have a look around at other lemon sorbet recipes, and see if you're using relatively more.

But if your recipe is sane, then it could be your process. Here's how it could mess things up.

Sorbet is basically sugar water with a bit of flavor. When it freezes all the way, unless it was churned as it froze, there's going to be a tendency for it to separate into ice crystals and even stronger sugar water. Some of this may happen in your first freezing step. Then you're pulverizing it in a food processor, probably giving it a chance to melt a bit. This is much worse than just leaving it out, since once it's in tiny bits, there's a lot more surface area exposed to air, and you're continuously mixing it, bringing in warm air, and so on. Then finally, it refreezes. The part that managed to melt in the meantime is going to repeat the earlier process, except now there's lots of space, so it's even easier for ice crystals to form and force out the sugar. So I'd definitely expect a tendency to end up with a runny sugar syrup component to your "sorbet". (If it's at all grainy, this is probably what's happening.)

If you're determined to do this without any kind of ice cream maker, it's going to be hard to avoid problems. You can certainly try to correct for it by using less sugar, though. You could also just make a granita - freeze it on a sheet, and scrape it off to serve.

There are a few things to consider here. Firstly, sugar dissolves in hot water better than in cold water. So, regardless of other considerations, it's worth making a simple syrup of the water and sugar.

What will change by heating is a few things. From a taste standpoint, depending on how you cook the fruit, the sorbet may taste "cooked". If that's the flavor you're looking for, cook away! However, "fresher" tasting sorbets are made by just pureeing (and straining). Cooking the fruit also stops oxidation (and discoloring) which happens with some fruits. On the other hand, the acidic lime juice will help prevent oxidation. Speaking of lime juice, if it's freshly-squeezed, a large component to its flavor are the volatile citrus oils. Cooking will likely cause these to evaporate, thereby losing some "citrusy" flavor (the acidity will remain, though).

From a texture standpoint it depends on a few things. Cooked fruit is usually softer; when pureed it will contribute towards a smoother texture. Additionally, many fruits contain pectin which is released when cooking. Pectin will contribute positively to the resulting texture of the sorbet. However, overcooking does degrade the pectin.

One way to get the best of both worlds is to gently poach the fruit in the simple syrup to soften it and release the pectin. (Alternatively, you can sous vide the fruit.) If you really want the "fresh" flavor (my preference) but want the benefit of pectin you can add a hydrocolloid thickener like xanthan gum or guar gum in a small amount (usually under 1% by weight).