To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote the number of lattes as \(x_1\) and the number of cappuccinos as \(x_2\).

The objective function is to maximize profit. Given that each latte provides $2 in profit and each cappuccino provides $1 in profit, we can write the objective function as:
\[ \text{Maximize:} \quad 2x_1 + x_2 \]

The constraints are based on the availability of milk and coffee. Since each latte requires 4 units of milk and each cappuccino requires 2 units of milk, and there are a total of 80 units of milk available, we have:
\[ 4x_1 + 2x_2 \leq 80 \]

For the coffee, since each latte needs 1 unit of coffee and each cappuccino needs 2 units of coffee, with a total of 50 units of coffee available, we get:
\[ x_1 + 2x_2 \leq 50 \]

Additionally, \(x_1\) and \(x_2\) must be non-negative since they represent the number of lattes and cappuccinos, respectively.

Thus, our symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of lattes'), ('x2', 'number of cappuccinos')],
    'objective_function': '2*x1 + x2',
    'constraints': ['4*x1 + 2*x2 <= 80', 'x1 + 2*x2 <= 50', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's implement this problem using Gurobi in Python:
```python
from gurobipy import *

# Create a new model
m = Model("Cafe_Optimization")

# Define the variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="lattes", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="cappuccinos", lb=0)

# Set the objective function
m.setObjective(2*x1 + x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(4*x1 + 2*x2 <= 80, "milk_constraint")
m.addConstr(x1 + 2*x2 <= 50, "coffee_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of lattes: {x1.x}")
    print(f"Number of cappuccinos: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```