To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables for the quantities of t-shirts and hoodies, formulating the objective function that represents the total profit, and listing all constraints based on the given limitations.

Let's denote:
- \(x_1\) as the number of t-shirts produced,
- \(x_2\) as the number of hoodies produced.

The objective function aims to maximize profit. Given that each t-shirt nets $10 in profit and each hoodie nets $15 in profit, the objective function can be written as:
\[ \text{Maximize: } 10x_1 + 15x_2 \]

Now, let's formulate the constraints based on the given information:
1. Designing time constraint: T-shirts require 1 hour of designing time, and hoodies require 2 hours of designing time. The designers are available for 40 hours a week.
\[ x_1 + 2x_2 \leq 40 \]

2. Printing time constraint: T-shirts require 2 hours of printing time, and hoodies require 3 hours of printing time. The printing machine is available for 60 hours per week.
\[ 2x_1 + 3x_2 \leq 60 \]

3. Non-negativity constraints: The number of t-shirts and hoodies cannot be negative.
\[ x_1 \geq 0, x_2 \geq 0 \]

In symbolic notation with natural language objects:
```json
{
    'sym_variables': [('x1', 'number of t-shirts'), ('x2', 'number of hoodies')],
    'objective_function': '10*x1 + 15*x2',
    'constraints': ['x1 + 2*x2 <= 40', '2*x1 + 3*x2 <= 60', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name="t-shirts", lb=0)
x2 = m.addVar(name="hoodies", lb=0)

# Set the objective function to maximize profit
m.setObjective(10*x1 + 15*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + 2*x2 <= 40, name="designing_time")
m.addConstr(2*x1 + 3*x2 <= 60, name="printing_time")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of t-shirts: {x1.x}")
    print(f"Number of hoodies: {x2.x}")
    print(f"Maximum profit: ${10*x1.x + 15*x2.x}")
else:
    print("No optimal solution found")
```